## Flexagons

Ever since my junior high math teacher taught me how to make flexagons, I’ve thought they were cool. Recently, I’ve started to make flexagons with pictures on the faces. I’ve sent several hexahexaflexagons as birthday cards. I’ve been experimenting with using a dodecahexaflexagon as a rosary or prayer wheel analog. A meditative tool for focusing the mind.

Flexagons were discovered in 1939 by Arthur H. Stone, then a graduate student at Princeton University. He and his colleagues developed much of what is known about flexagons. Most of what I know about flexagons came from Martin Gardner’s “Mathematical Games” columns in Scientific American, which are available in book form. I have borrowed the figure below, demonstrating how to flex, from his article. (The other figures on these pages are my creations.)

There are many kinds of flexagons, with varying numbers of faces and edges, but I find the symmetric hexaflexagons the most interesting. They are characterized by three-fold symmetry at the “center” with two-fold symmetry in the “outer” layers. They can be made with 3 times any power of two faces, that is, 3, 6, 12, 24, etc.

Though the higher order flexagons are difficult to make, due to the mechanical limitations of paper, they are easy to simulate on a computer.

### Making a Hexahexaflexagon

I have included instructions for making a hexahexaflexagon (6 faces) in a separate page. It is simple to generalize these to any higher order.

### How to Flex a Hexaflexagon

Flex a flexagon by pinching together two adjacent triangles (top left). The inner edge of the two opposite triangles may be opened with the other hand (top right to bottom). If the flexagon opens, it can be turned inside out, revealing a face that was not visible before. If it cannot be opened, then pinch an adjacent pair of triangles instead.

There’s an easy rhythm I fall into while flexing, where I flex and then rotate the flexagon 60 degrees, and then flex again and rotate. This flips through a cycle of the same three faces, over and over. This can be an aid to meditation in much the same way that a rosary or a prayer wheel can be.

Not rotating the flexagon between flexes, or rotating an extra 60 degrees allows escape from the original cycle into another, which includes the face just flexed from, plus the face which would have been flexed into, with its triangles permuted, plus a new face not part of the original cycle. Sometimes there is only one way to flex a face, indicating that escape will have to wait for another face.

### Finding All the Faces Of a Hexaflexagon (the Tuckerman Traverse)

Flexing the same edge repeatedly, rotating only when necessary to flex, eventually cycles through all the faces. This is known as a Tuckerman traverse, after Bryant Tuckerman, who first discovered it.

The faces are numbered such that faces 1, 2, and 3 are the faces of a trihexaflexagon, and adding faces 4, 5, and 6 yields the additional faces of a hexahexaflexagon, etc. That way, the face numbers remain consistent for a trihexaflexagon, a hexahexaflexagon, a dodecahexaflexagon, or any higher order.

State diagrams for different orders of hexaflexagons can be easily generalized from the diagram.

The state diagram below shows a Tuckerman traverse on a dodecahexaflexagon. Each circle represents a unique configuration of the flexagon. The numbers within each circle represent the top and bottom faces in that configuration. The arrows indicate the direction of traverse. Rotating and flexing is represented in the diagram by exiting a configuration via the other exit.

The various trianges in the diagram represent 3-cycles into which a flex-and-rotate combination can lead you. This flex-and-rotate pattern is very easy to do, as it is a natural movement to rotate the flexagon as you open it up from flexing.

Turning the flexagon over and flexing will run backwards through the diagram.

### An Interactive Flexagon Flexer

If your browser supports Java, you may wish to try my interactive hexaflexagon flexer, which I’ve put on a separate page. This example flexes a dodecahexaflexagon (12 faces) with face diagrams as in my examples on each face, so that you can see the permutations. There is also a flexer which has artwork on each face.

### Other Flexagon Pages

I should have included this several years ago, but I’ve been very lazy. I’ll try to add more, but at the moment, the most fun pages are Vi Hart’s flexagon videos. I think the easiest way to see them all is at her Khan Academy site.

1. fook h eng says:

do you know of any 12 face flexagons actually made?

my website is wumingzuo.com, but the only part that i think may share an interest with you is http://www.wumingzuo.com/prose/pi_bolero.html

your website appears interesting and i am an ancient being and sometimes can’t overcome the inertia in order to explore new worlds. really, check http://www.wumingzuo.com/fook_in_paris.html to verify.

• Yes, I make 12-face flexagons out of paper strips. It’s my preferred size.

I have a Java program that takes 12 image files, cuts them into triangles, and lays them out into strips. These strips are written into a new image file that can be printed, cut into the strips, and glued together. The endpoint triangles are duplicated so that they can be glued face-to-face.

The pi-bolero site was interesting! Thanks!

My father was in the Navy at the tail-end of WWII, but did not see combat. He seems to have been older than you…

2. Cannot keep in mind the identify of an old 80′s arcade puzzle game……it was like vector…?

3. Preeti says:

Hiiee
i want to try making a flexagon with three sides, but want to print my own pattern on it. Can you please help me do that.
It will be a great help :)

• I have a Java project at https://github.com/khuxtable/flexagon-layout that you may be able to use. I don’t have time (I’m very busy at work) to actually do a layout for you, but if you can build a Java project with Maven you may be able to use this. I reads JPG files and prints out a layout. I haven’t used it in years, though.

Another thing you can do is make a plain three sided flexagon, label everything, then disassemble it and use it as a guide.

4. nobo13 says:

Hi, I am wondering if you’ve heard or seen anyone who has made a 24 sided hexaflexagon? I was thinking of doing that with my class at Christmas

• Yes, I’ve made a 24-sided hexaflexagon. It’s very fragile, and very difficult to flex. But it can be done. I ended up cutting the corners a bit, which gave me a hole in the centre, and that helped a bit.